A wide variety of seismic modeling, processing and inversion algorithms require the recalculation of the seismic response after incremental local alterations to an initial seismic finite-difference model. For example, pre-stack finite-difference migration of seismic data provides a highly accurate means of producing images of the Earth's interior. The migration algorithm consists of recalculating the finite-difference response of small local changes to the seismic model. However, full finite-difference migration is rarely performed because of computational limitations restricting migration algorithms to the use of less accurate asymptotic techniques. Another example relates to finite-difference inversion, where recalculating the finite-difference response is the core (forward modeling step) of the algorithms.
Yet another example which is considered as being an important area of the present invention refers to so-called time-lapse seismics (or 4-D seismics). In this application it is of interest to investigate the effects that small (local) changes to the model have on the seismic response, e.g., varying water-oil-contact levels in a producing reservoir.
Also, in forward modeling, it may be of interest to re-compute the response of an altered seismic model. Forward modeling may serve as a means of learning what effects certain features of a seismic model have on the full response. Also, as the knowledge of the model evolves, or as it becomes more refined, a simulated response may need to be updated.
Another area of interest regarding the present invention lies in Amplitude Variation with Offset (AVO) calculations, where the effects of, for instance, changes of the degree of anisotropy of a cap-rock may be the target of investigation.
Furthermore, FD modeling has been used in connection with borehole measurements, simulations of tool behavior and characteristics in their operational environment. Typically, it is of interest to investigate the effects that small changes to the tool design or model parameters have on the propagation of waves in the vicinity of the tool.
The common feature of these problems is that changes to the model are often restricted to a small sub-volume, but finite-difference simulations are required for the full model with several alterations. A method that would allow full finite-difference simulations for the complete model to be corrected for these changes while only requiring calculations in the sub-volume and its neighborhood could significantly reduce the computational cost both in terms of the number of calculations and memory for storage of material parameters and variable fields.
Finite-difference methods provide an accurate way of computing seismograms from complex seismic models. However, as mentioned above, the finite-difference simulations tend to become prohibitively expensive to run on even state-of-the-art computing equipment. Therefore, different approaches have been taken to make highly accurate numerical modeling methods such as finite-difference schemes more efficient. Two major directions of effort to achieve significant computational savings can be found in the literature: (1) hybrid techniques; and (2) grid-refinement techniques.
By combining methods appropriate for different wave propagation regimes, it is possible to increase computational efficiency as well as the simulation accuracy considerably. For details of such an approach, reference is made to Wu, R. S. and R. Aki, 1988, Introduction: Seismic wave scattering in three-dimensionally heterogeneous Earth, in: Scattering and Attenuation of Seismic Waves, edited by K. Aki and R. S. Wu, pp. 1-6. Birkhauser Verlag, Basel, Switzerland. Several such hybrid techniques have been developed for seismic applications. For example, a ray method can be used to propagate energy over long distances into an acoustic finite-difference grid at the scattering site. Also described are elastic methods combining boundary integral and finite-difference techniques. The hybridization, i.e. the interchange of wave fields between the methods, is based on Green's theorem. Stead and Helmberger (Stead, R. J. and D. V. Helmberger, 1988, Numerical-analytical interfacing in two dimensions with applications to modeling NTS seismograms, in: Scattering and Attenuation of Seismic Waves, see above, pp. 157-193) achieved the numerical propagation of an elastic wave field by means of the Kirchhoff integral by partitioning the wave field into separate compressional and shear components before integration.
Emmerich (Emmerich, H., PSV-wave propagation in a medium with local heterogeneities: a hybrid formulation and its application, Geophys. J. Int. 109, 54-64 (1992)) combined a reflectivity solution with a viscoelastic finite-difference scheme. The technique is efficient because it assumes that the scattering targets are overlain by stratified media. A hybrid technique can also be based on a reflectivity and a pseudo-spectral scheme to solve anelastic scattering problems.
The use of hybrid-FD schemes in borehole seimics is described by Kurkjian, A. L., R. T. Coates, J. E. White and H. Schmidt, Finite-difference and frequency-wavenumber modeling of seismic monopole sources and receivers in fluid-filled boreholes, Geophysics 59(1994), 1053-1064. The authors model sources and receivers in the presence of boreholes by interfacing a frequency-wavenumber method with a finite-difference scheme.
Robertsson et al. introduced an integrated Gaussian-beam technique with viscoelastic finite differences and a Kirchhoff method to simulate deep ocean seafloor scattering experiments (a hybrid technique referred to as HARVEST (Hybrid Adaptive Regime Visco-Elastic Simulation Technique), see Robertsson, J. O. A., J. O. Blanch and W. W. Symes, Viscoelastic finite-difference modeling, Geophysics 59(1994), 1444-1456; Robertsson, J. O. A., A. Levander and K. Holliger, Modeling of the Acoustic Reverberation Special Research Program deep ocean seafloor scattering experiments using a hybrid wave propagation simulation technique, J. Geophys. Res. 101(1996), 3085-3101 and (by the same authors) A hybrid wave propagation simulation technique for ocean acoustic problems, in: J. Geophys. Res. 101(1996), 11225-11241.
In inserting a wave field inside a finite-difference grid, there have generally been two approaches taken. Either, as was described by Alterman, Z. and F. C. Karal, Propagation of elastic waves in layered media by finite difference methods, Bull. Seis. Soc. Am. 58(1968) 367-398, the source wave field is inserted along a closed boundary inside the finite-difference grid so that the source wave field radiates out from it into the main part of the grid. The other approach has been to insert the wave field along a line inside a finite-difference grid which leads to artificial edge diffractions but allows coupling of different simulation methods (see for example Robertsson et al., J. Geophys. Res. 101(1996), 11225-11241).
Based on the above, Zahradnik, J. and P. Moczo, in: Hybrid seismic modeling based on discrete-wave number and finite-difference methods, Pure Appl. Geophys.148(1996), 21-38, inserted the source wave field along an open boundary bounded by a free surface (reflecting) at one side. The source wave field is calculated by an FD method using a homogeneous background medium. The FD calculation was repeated within the open boundaries after introducing a shallow basin into the homogeneous background. While including steps also found in the present invention, Zahradnik and Moczo used the known method solely to verify a hybrid (DW-FD) technique for earthquake seismology.
With respect to the second major direction of efforts to achieve significant computational savings, i.e., grid-refinement techniques, there are two constraints which limit finite-difference calculations. Those are the shortest wavelengths that occur in the simulation model and the complexity of the model. A maximum grid-step size to achieve a sufficiently accurate solution is constrained by either of these two conditions. By applying a finer grid in the parts of the model where the lowest seismic velocities or the highest structural complexity occur, a computationally efficient solution may be obtained. The complicated issue here is how to connect the different finite-difference grids to each other without introducing artificial boundary reflections.
McLaughlin, K. L. and S. M. Day in: 3D elastic finite-difference seismic-wave simulations, Computers in Physics 8(1994), 656-663, describe a three-dimensional (3-D) finite-difference grid-refinement technique based on viewing the entire grid as a tree structure of sub-grids with various discretizations.
In view of the above cited prior art it is an object of the invention to provide methods for improving the efficiency of seismic wave-field calculation. It is a more specific object of the invention to improve the efficiency and applicability of finite-difference methods for seismic exploration techniques, particularly for techniques which employ models subject to alteration(s).